Problem 15 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 18 cups to 45 cups

Step-by-Step Solution

Answer

The ratio 18 cups to 45 cups as a fraction in simplest form is \( \frac{2}{5} \).

Step 1: Write the Ratio as a Fraction

First, express the given ratio of 18 cups to 45 cups as a fraction where the first number is the numerator and the second is the denominator. Thus, the fraction becomes \( \frac{18}{45} \).

Step 2: Find the Greatest Common Divisor (GCD)

The next step is to find the greatest common divisor of 18 and 45. List the factors of both numbers: - Factors of 18: 1, 2, 3, 6, 9, 18 - Factors of 45: 1, 3, 5, 9, 15, 45 The greatest common divisor is 9.

Step 3: Simplify the Fraction

To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 9. \[ \frac{18 \div 9}{45 \div 9} = \frac{2}{5} \]

Step 4: Confirm the Simplified Fraction

Finally, ensure that \( \frac{2}{5} \) is in its simplest form. Since both numbers in the fraction are prime relative to each other (having no common factors other than 1), \( \frac{2}{5} \) is indeed the simplest form.

Key Concepts

Ratios

A ratio is a comparison between two quantities. It tells us how many times one number contains another. In the exercise, we have a ratio of 18 cups to 45 cups.

• Ratios can appear in different forms, such as '18 to 45', or as a fraction \( \frac{18}{45} \).
• This gives a clear picture of how one quantity measures up to the other.

Understanding ratios is very important, especially in everyday situations like recipes, where you need to mix ingredients in right proportions. Ratios can be converted to fractions to make it easier to simplify or work with them in mathematical equations.

Fractions

Fractions are a way to represent numbers that are not whole. A fraction consists of a numerator, the top number, and a denominator, the bottom number.

• When converting a ratio like '18 cups to 45 cups' to a fraction, 18 becomes the numerator, and 45 becomes the denominator, resulting in \( \frac{18}{45} \).
• Fractions allow us to easily perform mathematical operations like addition, subtraction, multiplication, and division.

They are flexible and can be simplified for easier computation, which leads us to the next concept.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial for simplifying fractions. The GCD of two numbers is the largest number that divides both numbers evenly.

• To find the GCD of 18 and 45, we list out their factors: 18 is divisible by 1, 2, 3, 6, 9, and 18, while 45 is divisible by 1, 3, 5, 9, 15, and 45.
• The largest common factor is 9, making it the GCD.

By using the GCD, we can effectively reduce our fraction \( \frac{18}{45} \) to its simplest form, ensuring it's as easy to understand as possible.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This form makes fractions easier to work with and understand.

• To simplify \( \frac{18}{45} \), divide both the numerator and the denominator by their greatest common divisor, which is 9.
• Performing the division, we get \( \frac{2}{5} \), which is the simplest form.

Simplification is helpful because it allows us to see the true relationship between quantities without unnecessary complexity. For example, instead of thinking about the relationship as \( \frac{18}{45} \), we think of it as \( \frac{2}{5} \), which is often much clearer.